This note is interesting in that there is only one owner named on it. Were there many Private Banks with just one partner, or am I missing something?
The design of the note is similar to notes of a bank of the same name but with different partners named on its notes.
I have to see if I can find a pic of one of the other notes.

I have seen other examples where just one partner is listed. The legal maximum was 6 partners but I am not aware of a restriction requiring more than one partner. The odd denomination relates to the fact that it is equivalent to 1/6th of a Guinea. This denomination was issued to compensate for the lack of silver coinage and such notes were known as Silver Notes.

There was also a Silver Bank based in Malahide with partners, Talbot and Glascock. I attach an example of their silver note.